Solver Finding the Equation of a Line

Source code of 'Finding the Equation of a Line'

This Solver (Finding the Equation of a Line) was created by by jim_thompson5910(35256)

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About jim_thompson5910: If you need more math help, then you can email me. I charge $2 for steps, or $1 for answers only. Email: jim_thompson5910@hotmail.com Website: http://www.freewebs.com/jimthompson5910/home.html

==section input
Input two points {{{x[1]}}}=*[input x_coord1=-4] {{{y[1]}}}=*[input y_coord1=-8] {{{x[2]}}}=*[input x_coord2=-3] {{{y[2]}}}=*[input y_coord2=1] to find the slope and the equation of the line

Disclaimer: This solver may not be perfect. If there are any errors, please report them to me (please be respectful when reporting the errors). Please make sure to follow all instructions the automated solver provides. Always keep this in mind: an automated solver like this is NOT a suitable replacement for your brain. So please don't blindly follow the solver completely. It's simply a tool like a calculator is. You shouldn't blindly trust your calculator 100% of the time. If something doesn't look right, seek a second opinion and let me know about it. Finally, make sure that you're using the correct solver in the correct way. For example, if you need to find the equation of the line, don't use the "distance formula solver" even if there are some overlapping ideas. The solvers won't have any clue about what you truly mean.

==section solution

perl

$point1="$x_coord1,$y_coord1";
$point2="$x_coord2,$y_coord2";

sub check_denom
{my $input=$_[0];
 if($input==0)
   {return undef;}
}

sub strip
{
my (@input_array);
@input_array=@_;
if($input_array[1]=~m/\//)
  {$input_array[1]=$input_array[2];
   pop(@input_array);}
return @input_array;}

sub mult{

my ($split,@arr1,@arr2,$temp1,$temp2, @hold);

@hold=@_;
if($hold[1]=~m/\//)
{@arr1=($hold[0],$hold[2]);}
if($hold[4]=~m/\//)
{@arr2=($hold[3],$hold[5]);}

if($hold[1]!~m/\//)
{@arr1=($hold[0],$hold[1]);}
if($hold[3]!~m/\//)
{@arr2=($hold[2],$hold[3]);}

if($arr1[1]=~m/\//)
{@arr1=strip(@arr1);}
if($arr2[1]=~m/\//)
{@arr2=strip(@arr2);}

#print "arr1=@arr1\n\n";

$temp1=$arr1[0]*$arr2[0];
$temp2=$arr1[1]*$arr2[1];

#$temp1=$numer1*$numer2;
#$temp2=$denom1*$denom2;
return ($temp1,"/",$temp2);
}

sub add{

my ($split,@arr1,@arr2, @hold, @tempfrac, $gcf, @tempor_frac);

@hold=@_;
if($hold[1]=~m/\//)
{@arr1=($hold[0],$hold[2]);}
if($hold[4]=~m/\//)
{@arr2=($hold[3],$hold[5]);}

if($hold[1]!~m/\//)
{@arr1=($hold[0],$hold[1]);}
if($hold[3]!~m/\//)
{@arr2=($hold[2],$hold[3]);}

if($arr1[1]=~m/\//)
{@arr1=strip(@arr1);}
if($arr2[1]=~m/\//)
{@arr2=strip(@arr2);}

if(($arr1[1]!=$arr2[1])&&(($arr1[1]!=0)&&($arr2[1]!=0)))
{
 for(my $count=1;$count<abs($arr1[1]*$arr2[1]);$count++)
  {if(($arr1[1]%$count==0)&&($arr2[1]%$count==0))
    {$gcf=$count;}
  }

my $lcm=($arr1[1]*$arr2[1])/$gcf;
my $temp=$lcm/$arr1[1];

$exported_lcm=$lcm;
$exported_gcf=$gcf;

$arr1[0]=($lcm/$arr1[1])*$arr1[0];
$arr2[0]=($lcm/$arr2[1])*$arr2[0];

$arr1[1]=($lcm/$arr1[1])*$arr1[1];
$arr2[1]=($lcm/$arr2[1])*$arr2[1];
}
$tempfrac1[0]=$arr1[0]+$arr2[0];
$tempfrac2[0]=$arr1[1];
return ($tempfrac1[0],"/",$tempfrac2[0]);
}

sub subtract{

my ($split,@arr1,@arr2, @hold, @tempfrac, $gcf);

@hold=@_;
@arr1=($hold[0],$hold[2]);
@arr2=($hold[3],$hold[5]);

@arr1=strip(@arr1);
@arr2=strip(@arr2);

if(($arr1[1]!=$arr2[1])&&(($arr1[1]!=0)&&($arr2[1]!=0)))
{
 for($count=1;$count<abs($arr1[1]*$arr2[1]);$count++)
  {if(($arr1[1]%$count==0)&&($arr2[1]%$count==0))
    {$gcf=$count;}
  }
my $lcm=($arr1[1]*$arr2[1])/$gcf;
my $temp=$lcm/$arr1[1];

$exported_lcm=$lcm;
$exported_gcf=$gcf;

$arr1[0]=($lcm/$arr1[1])*$arr1[0];
$arr2[0]=($lcm/$arr2[1])*$arr2[0];

$arr1[1]=($lcm/$arr1[1])*$arr1[1];
$arr2[1]=($lcm/$arr2[1])*$arr2[1];
}
$tempfrac1[0]=$arr1[0]-$arr2[0];
$tempfrac2[0]=$arr1[1];
return ($tempfrac1[0],"/",$tempfrac2[0]);
}

sub reduce{

my (@arr1, @original_arr1, $flag);

$exported_no_reduce=0;
@arr1=@_;

if($arr1[1]=~m/\//)
{@arr1=strip(@arr1);}

@original_arr1=@arr1;

$flag=$_[-1];

if($arr1[0]==0)
{return $arr1[0];}

if($arr1[1]!=0)
{
$store=$arr1[0]/$arr1[1];

if($store>0)
{
  $arr1[0]=abs($arr1[0]);
  $arr1[1]=abs($arr1[1]);
}

if(($store<0)&&($arr1[1]<0))
{
  $arr1[0]=0-$arr1[0];
  $arr1[1]=abs($arr1[1]);
}

if($arr1[0]%$arr1[1]==0)
{
  #if($flag==1)
   #{$arr1[0]=$arr1[0]/$arr1[1];
   #return $arr1[0];}

if($flag==2)
   {$arr1[0]=$arr1[0]/$arr1[1];
   $arr1[1]=1;
   return ($arr1[0],"/",$arr1[1]);}
#  if($flag==0)
#   {return $quotient;}

}

for (my $i=abs($arr1[0]*$arr1[1]); $i>1; $i--)
{
 if (($arr1[0]%$i==0)&&($arr1[1]%$i==0))
    {
       $arr1[0]/=$i;
       $arr1[1]/=$i;
       my $j=$i;

}
}

if($original_arr1[0]==$arr1[0])
{$exported_no_reduce=1;}

if($original_arr1[0]==-$arr1[0])
{$exported_no_reduce=2;}

return ($arr1[0],"/",$arr1[1]);
}

sub invert{

my(@arr1,$temp);

@arr1=@_;

@arr1=strip(@arr1);

$temp=$arr1[0];
$arr1[0]=$arr1[1];
$arr1[1]=$temp;

return ($arr1[0],"/", $arr1[1]);
}

sub divide{

my ($split,@arr1,@arr2,$temp1,$temp2, @hold);

@hold=@_;
@arr1=($hold[0],$hold[2]);
@arr2=($hold[3],$hold[5]);

@arr1=strip(@arr1);
@arr2=strip(@arr2);

#print "arr1=@arr1\n\n";

$temp1=$arr1[0]*$arr2[1];
$temp2=$arr1[1]*$arr2[0];

#$temp1=$numer1*$numer2;
#$temp2=$denom1*$denom2;
return ($temp1,"/",$temp2);
}

sub clean
{ my ($string_clean,$original_string);
$string_change=0;
$string_clean=$_[0];
$original_string=$string_clean;
$string_clean=~s/\s//g;
$string_clean=~s/\/1\D//g;
$string_clean=~s/\/1$//;
$string_clean=~s/^\+//g;
$string_clean=~s/\W0\/\d+/0/g;
$string_clean=~s/\+\-/\-/g;
$string_clean=~s/\-\-/\+/g;
$string_clean=~s/^$(\d+)$/\1/;
$string_clean=~s/^$(\-\d+)$/\1/;

$string_clean=~s/$(\d+)$x/\1x/;
$string_clean=~s/$(\-\d+)$x/\1x/;

$string_clean=~s/\/(\D)/\1/;
$string_clean=~s/\/$//;

$string_clean=~s/\(\-0/\(0/g;

$string_clean=~s/(\+|\-)1{1}([a-z])/$1$2/;

if($original_string ne $string_clean)
{$string_change=1;}

return $string_clean;
}

@split_point1=split(/,/,$point1);
#$x_coord1=$split_point1[0];
$x_string=$x_coord1;
#$y_coord1=$split_point1[1];
$y_string=$y_coord1;

@split_point2=split(/,/,$point2);
#$x_coord2=$split_point2[0];
#$y_coord2=$split_point2[1];

$offset=9;

$rad=(abs(($x_coord+$offset)-($x_coord-$offset)))/150;

$x_avg=($x_coord1+$x_coord2)/2;
$y_avg=($y_coord1+$y_coord2)/2;

$graph_dimensions="500, 500, ".($x_avg-$offset).", ".($x_avg+$offset).", ".($y_avg-$offset).", ".($y_avg+$offset)."";

#########

###x-coord1
if($x_coord1=~/\//)
{@x_coord1=split(/\//,$x_coord1);

$x_coord1[2]=$x_coord1[1];
$x_coord1[1]="/";
$x_is_fraction1=1;}

if($x_coord1!~/\//)
{$x_coord1[0]=$x_coord1;
$x_coord1[1]="/";
$x_coord1[2]=1;}
###

###y-coord1
if($y_coord1=~/\//)
{@y_coord1=split(/\//,$y_coord1);

$y_coord1[2]=$y_coord1[1];
$y_coord1[1]="/";
$y_is_fraction1=1;}

if($y_coord1!~/\//)
{$y_coord1[0]=$y_coord1;
$y_coord1[1]="/";
$y_coord1[2]=1;}
###

###x-coord2
if($x_coord2=~/\//)
{@x_coord2=split(/\//,$x_coord2);

$x_coord2[2]=$x_coord2[1];
$x_coord2[1]="/";
$x_is_fraction2=1;}

if($x_coord2!~/\//)
{$x_coord2[0]=$x_coord2;
$x_coord2[1]="/";
$x_coord2[2]=1;}
###

###y-coord2
if($y_coord2=~/\//)
{@y_coord2=split(/\//,$y_coord2);

$y_coord2[2]=$y_coord2[1];
$y_coord2[1]="/";
$y_is_fraction2=1;}

if($y_coord2!~/\//)
{$y_coord2[0]=$y_coord2;
$y_coord2[1]="/";
$y_coord2[2]=1;}
###

###########Now lets use the  Point-Slope Formula to find the equation of the line

print "First lets find the slope through the points ({{{$x_coord1}}},{{{$y_coord1}}}) and ({{{$x_coord2}}},{{{$y_coord2}}})

{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula (note: ({{{x[1]}}},{{{y[1]}}}) is the first point ({{{$x_coord1}}},{{{$y_coord1}}}) and  ({{{x[2]}}},{{{y[2]}}}) is the second point ({{{$x_coord2}}},{{{$y_coord2}}}))

{{{m=($y_coord2-$y_coord1)/($x_coord2-$x_coord1)}}} Plug in {{{y[2]=$y_coord2}}},{{{y[1]=$y_coord1}}},{{{x[2]=$x_coord2}}},{{{x[1]=$x_coord1}}}  (these are the coordinates of given points)

if(($x_coord1[2]==1)&&($y_coord1[2]==1))
{
 if(($x_coord2[2]==1)&&($y_coord2[2]==1))
  {

$rise=$y_coord2-$y_coord1;
  $run=$x_coord2-$x_coord1;
  @slope=reduce($rise,$run);
  $slope_string=clean("@slope");

print "{{{m= $rise/$run}}} Subtract the terms in the numerator {{{$y_coord2-$y_coord1}}} to get {{{$rise}}}.  Subtract the terms in the denominator {{{$x_coord2-$x_coord1}}} to get {{{$run}}}

if($exported_no_reduce!=1)
  {print "

{{{m=$slope_string}}} Reduce

";}
 }

}
else
{@rise_array=add($y_coord2[0],$y_coord2[2],-$y_coord1[0],$y_coord1[2]);
 @run_array=add($x_coord2[0],$x_coord2[2],-$x_coord1[0],$x_coord1[2]);
 $rise="@rise_array";
 $run="@run_array";
 @slope=divide(@rise_array,@run_array);
 @slope=reduce(@slope);
 $slope_string="@slope";

print "{{{m= ($rise)/($run)}}} Subtract   (note: if you need help with subtracting or dividing fractions, check out this <a href=http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver>solver</a>)

{{{m=@slope}}} Divide the fractions

";
}

if($x_coord1==$x_coord2)
{print "

Since the denominator is zero, the slope is undefined (remember you cannot divide by zero). So we cannot use the slope intercept form to write an equation. So we can only say that the equation is a vertical line through {{{x=$x_coord1}}}, which means the equation is {{{x=$x_coord1}}} (notice this is <font size=4><b>not</b></font> in slope-intercept form)

So the equation {{{x=$x_coord1}}} looks like this:

{{{drawing($graph_dimensions,
graph($graph_dimensions,1000(x-$x_coord1)),
circle($point1,0.1),
circle($point1,0.1+0.03),
circle($point2,0.1),
circle($point2,0.1+0.03)
) }}} Graph of {{{x=$x_coord1}}} through the points ({{{$x_coord1}}},{{{$y_coord1}}}) and ({{{$x_coord2}}},{{{$y_coord2}}})

return;
}

print "

So the slope is

{{{m=$slope_string}}}

------------------------------------------------

#If you want to find the equation of line with through the points ({{{$x_coord1}}},{{{$y_coord1}}}) and ({{{$x_coord2}}},{{{$y_coord2}}}), you can simply

print "\n\nNow let's use the point-slope formula to find the equation of the line:

------Point-Slope Formula------
{{{y-y[1]=m(x-x[1])}}} where {{{m}}} is the slope, and ({{{x[1]}}},{{{y[1]}}}) is one of the given points

So lets use the Point-Slope Formula to find the equation of the line

{{{y-$y_string=($slope_string)(x-$x_string)}}} Plug in {{{m=$slope_string}}}, {{{x[1]=$x_string}}}, and {{{y[1]=$y_string}}} (these values are given)

$y_simp=clean("y-$y_string");

if($string_change==1)
{
print "

{{{$y_simp=($slope_string)(x-$x_string)}}} Rewrite {{{y-$y_string}}} as {{{$y_simp}}}

";
}

$x_simp=clean("x-$x_string");

if($string_change==1)
{
print "

{{{$y_simp=($slope_string)($x_simp)}}} Rewrite {{{x-$x_string}}} as {{{$x_simp}}}

";
}

if($x_simp=~m/\+/)
{substr($x_string,0,1)="";}
if($x_simp=~m/\-/)
{$x_string="-"."$x_string";}

$expression=clean("y-$y_string=($slope_string)x+($slope_string)($x_string)");

# to {{{$x_simp}}}

print "

{{{$expression}}} Distribute {{{$slope_string}}}
";

@temp_b=mult(-$x_coord1[0],$x_coord1[2],$slope[0],$slope[2]);
$temp_b_string_not_reduce=clean("@temp_b");
$storage="@temp_b";

$temp_b_string_not_reduce="@temp_b";
@temp_b=reduce(@temp_b);
$temp_b_string=clean("@temp_b");
#$temp_b_string=~s/\/1$//;

$expression=clean("y-$y_string=($slope_string)x+$temp_b_string");

$x_string=~s/\-0/0/;

print "

{{{$expression}}} Multiply {{{$slope_string}}} and {{{$x_string}}} to get {{{$temp_b_string_not_reduce}}}", ($temp_b_string_not_reduce ne $temp_b_string) ? ". Now reduce {{{$temp_b_string_not_reduce}}} to get {{{$temp_b_string}}}\n":"\n";

@abs=(abs($y_coord1[0]),"/",$y_coord1[2]);
$abs=clean("@abs");

$expression=clean("y=($slope_string)x+$temp_b_string+$y_string");

print "

{{{$expression}}} ", ($y_coord1[0]<0) ? "Subtract {{{$abs}}} from ":"Add {{{$abs}}} to ";
print " both sides to isolate y";

@temp_c=add($temp_b[0],$temp_b[2],$y_coord1[0],$y_coord1[2]);
@temp_c=reduce(@temp_c,1);
$temp_c_string="@temp_c";
$temp_c_string=clean($temp_c_string);
#$temp_c_string=~s/\s//g;
#$temp_c_string=~s/\/1$//;

$expression=clean("y=($slope_string)x+$temp_c_string");

print "

{{{$expression}}} Combine like terms {{{$temp_b_string}}} and {{{$y_string}}} to get {{{$temp_c_string}}} ";

if(($temp_b_string=~m/\//)||($y_string=~m/\//))
{print "(note: if you need help with combining fractions, check out this <a href=http://www.algebra.com/algebra/homework/NumericFractions/fractions-solver.solver>solver</a>)

";}

$final=clean("($slope_string)x+$temp_c_string");

$real_slope=$slope_string;
$y1_string=$temp_c_string;

#return;

print "

------------------------------------------------------------------------------------------------------------

Answer:

So the equation of the line which goes through the points ({{{$x_coord1}}},{{{$y_coord1}}}) and ({{{$x_coord2}}},{{{$y_coord2}}})  is:{{{y=$final}}}

The equation is now in {{{y=mx+b}}} form (which is slope-intercept form) where the slope is {{{m=$real_slope}}} and the y-intercept is {{{b=$y1_string}}}

Notice if we graph the equation {{{y=$final}}} and plot the points ({{{$x_coord1}}},{{{$y_coord1}}}) and ({{{$x_coord2}}},{{{$y_coord2}}}),  we get this: $note: if you need help with graphing, check out this \<a href=http\:/\/www\.algebra\.com\/algebra\/homework\/Linear-equations\/graphing\-linear\-equations\.solver\>solver\<a\>$

{{{drawing($graph_dimensions,
graph($graph_dimensions,($real_slope)x+$y1_string),
circle($point1,$rad),
circle($point1,$rad+0.03),
circle($point2,$rad),
circle($point2,$rad+0.03)
) }}} Graph of {{{y=$final}}} through the points ({{{$x_coord1}}},{{{$y_coord1}}}) and ({{{$x_coord2}}},{{{$y_coord2}}})

Notice how the two points lie on the line. This graphically verifies our answer.

#and we can see that the point lies on the line. Since we know the equation has a slope of {{{$real_slope}}} and goes through the #point ({{{$final_x_string}}},{{{$final_y_string}}}), this verifies our answer.";

==section output

==section check